# 一维无限长谐振子模型

$y(t)=Acos(\omega t+\phi_0)$

$y(x,t)=Acos(kx-\omega t+\phi_0)$

$\left|\psi\right>=\psi(x,t)=Ae^{i(kx-\omega t+\phi_0)}=Acos(kx-\omega t+\phi_0)+iAsin(kx-\omega t+\phi_0)$

# 薛定谔方程的启发式推导

$\frac{\partial \psi}{\partial t}=-i\omega\psi,\frac{\partial \psi}{\partial x}=ik\psi,\frac{\partial^2\psi}{\partial x^2}=-k^2\psi$

$i\hbar\frac{\partial\psi}{\partial t}=\hbar\omega\psi=E_{wave}\psi$

$p\psi=\hbar k\psi=-i\hbar\frac{\partial}{\partial x}\psi\\ p^2\psi=\hbar^2 k^2\psi=-\hbar^2\frac{\partial^2}{\partial x^2}\psi\\ i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\nabla^2\psi+U(x)\psi$

# 量子力学算符

$\hat{p}=-i\hbar\frac{\partial}{\partial x}$

\begin{align*} [x,p]f(x)&=x\hat{p}f(x)-\hat{p}xf(x)\\ &=-i\hbar\frac{\partial f(x)}{\partial x}+i\hbar\frac{\partial(xf(x))}{\partial x}\\ &=-i\hbar\frac{\partial f(x)}{\partial x}+i\hbar f(x)+i\hbar\frac{\partial f(x)}{\partial x}\\ &=i\hbar f(x)\Rightarrow[x,p]=i\hbar \end{align*}

$H=\frac{p^2}{2m}+U(x)\Rightarrow\hat{H}=-\frac{\hbar^2}{2m}\nabla^2+U(x)=i\hbar\frac{\partial}{\partial t}$

$\frac{\partial H}{\partial p}=\frac{p}{m}=\dot{x},\frac{\partial H}{\partial x}=\frac{\partial U(x)}{\partial x}=-F=-m\ddot{x}=-\dot{p}$

# 通用电子结构问题与BO近似

$H_{total}=\sum_{i=1}^K\frac{p_i^2}{2M_i}+\sum_{i=1}^N\frac{p_i^2}{2m_e}-\sum_{i,j}\frac{e^2}{4\pi\epsilon_0}\frac{Z_j}{\left|r_i-R_j\right|}+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{4\pi\epsilon_0}\frac{1}{\left|r_i-r_j\right|}+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{4\pi\epsilon_0}\frac{Z_iZ_j}{\left|r_i-r_j\right|}$

$\psi(r,R)=\chi(R)\psi(R;r)$

$H(p,r)=\sum_{i=1}^N\frac{p_i^2}{2m_e}-\sum_{i,j}\frac{e^2}{4\pi\epsilon_0}\frac{Z_j}{\left|r_i-R_j\right|}+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{4\pi\epsilon_0}\frac{1}{\left|r_i-r_j\right|}$

# 一次量子化

$H=-\sum_i\frac{\nabla^2_i}{2}-\sum_{i,j}\frac{Z_j}{\left|r_i-R_j\right|}+\frac{1}{2}\sum_{i\neq j}\frac{1}{\left|r_i-r_j\right|}$

$\psi(\bold{x_0},…,\bold{x_{N-1}})= \frac{1}{\sqrt{N!}}\left|\begin{matrix} \phi_0(\bold{x_0})&\phi_1(\bold{x_0})&…&\psi_{M-1}(\bold{x_0})\\ \phi_0(\bold{x_1})&\phi_1(\bold{x_1})&…&\psi_{M-1}(\bold{x_1})\\ .&.&.&.\\ .&.&.&.\\ .&.&.&.\\ \phi_0(\bold{x_{N-1}})&\phi_1(\bold{x_{N-1}})&…&\psi_{M-1}(\bold{x_{N-1}}) \end{matrix} \right|$

$\psi(\bold{x_0},\bold{x_1},…,\bold{x_{N-1}})=-\psi(\bold{x_1},\bold{x_0},…,\bold{x_{N-1}})$

# 二次量子化

$H=\sum_{p,q}h_{pq}a^{\dagger}_pa_q+\frac{1}{2}\sum_{p,q,r,s}h_{pqrs}a^{\dagger}_pq^{\dagger}_qa_ra_s$

$h_{pq}=\left<\phi_p\right|\left(-\sum_i\frac{\nabla^2_i}{2}-\sum_{j}\frac{Z_j}{\left|r-R_j\right|}\right)\left|\phi_q\right>= \int d\bold{x}\phi_p^*(\bold{x})\left(-\sum_i\frac{\nabla^2_i}{2}-\sum_{j}\frac{Z_j}{\left|r-R_j\right|}\right)\phi_q(\bold{x})\\ h_{pqrs}=\left<\phi_p\right|\left<\phi_q\right|\frac{1}{\left|r_i-r_j\right|}\left|\phi_r\right>\left|\phi_s\right>= \int d\bold{x_1}d\bold{x_2}\frac{\phi_p^*(\bold{x_1})\phi_q^*(\bold{x_2})\phi_r(\bold{x_2})\phi_s(\bold{x_1})}{\left|r_i-r_j\right|}$

# 参考链接

1. https://zhuanlan.zhihu.com/p/139133715
2. Quantum Computational Chemistry. Sam McArdle, Suguru Endo and other co-authors.
3. https://arxiv.org/abs/2109.02110v1
原文作者：DECHIN
原文地址: https://www.cnblogs.com/dechinphy/p/second-quantization.html
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